(0) Obligation:
Clauses:
select(X1, [], X2) :- ','(!, failure(a)).
select(X, Y, Zs) :- ','(head(Y, X), tail(Y, Zs)).
select(X, Y, .(H, Zs)) :- ','(head(Y, H), ','(tail(Y, T), select(X, T, Zs))).
head([], X3).
head(.(H, X4), H).
tail([], []).
tail(.(X5, T), T).
failure(b).
Query: select(g,g,a)
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(2) Obligation:
Triples:
selectA(X1, .(X2, X3), .(X2, X4)) :- selectA(X1, X3, X4).
Clauses:
selectcA(X1, .(X1, X2), X2).
selectcA(X1, .(X2, X3), .(X2, X4)) :- selectcA(X1, X3, X4).
Afs:
selectA(x1, x2, x3) = selectA(x1, x2)
(3) TriplesToPiDPProof (SOUND transformation)
We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
selectA_in: (b,b,f)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
SELECTA_IN_GGA(X1, .(X2, X3), .(X2, X4)) → U1_GGA(X1, X2, X3, X4, selectA_in_gga(X1, X3, X4))
SELECTA_IN_GGA(X1, .(X2, X3), .(X2, X4)) → SELECTA_IN_GGA(X1, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
selectA_in_gga(
x1,
x2,
x3) =
selectA_in_gga(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
SELECTA_IN_GGA(
x1,
x2,
x3) =
SELECTA_IN_GGA(
x1,
x2)
U1_GGA(
x1,
x2,
x3,
x4,
x5) =
U1_GGA(
x1,
x2,
x3,
x5)
We have to consider all (P,R,Pi)-chains
Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
(4) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
SELECTA_IN_GGA(X1, .(X2, X3), .(X2, X4)) → U1_GGA(X1, X2, X3, X4, selectA_in_gga(X1, X3, X4))
SELECTA_IN_GGA(X1, .(X2, X3), .(X2, X4)) → SELECTA_IN_GGA(X1, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
selectA_in_gga(
x1,
x2,
x3) =
selectA_in_gga(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
SELECTA_IN_GGA(
x1,
x2,
x3) =
SELECTA_IN_GGA(
x1,
x2)
U1_GGA(
x1,
x2,
x3,
x4,
x5) =
U1_GGA(
x1,
x2,
x3,
x5)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
SELECTA_IN_GGA(X1, .(X2, X3), .(X2, X4)) → SELECTA_IN_GGA(X1, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
SELECTA_IN_GGA(
x1,
x2,
x3) =
SELECTA_IN_GGA(
x1,
x2)
We have to consider all (P,R,Pi)-chains
(7) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
SELECTA_IN_GGA(X1, .(X2, X3)) → SELECTA_IN_GGA(X1, X3)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(9) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- SELECTA_IN_GGA(X1, .(X2, X3)) → SELECTA_IN_GGA(X1, X3)
The graph contains the following edges 1 >= 1, 2 > 2
(10) YES